Hopping transport on site-disordered<i>d</i>-dimensional lattices

Abstract

We consider here the hopping of an electron among a band of localized electronic states on a d-dimensional lattice. The hopping rates are assumed to be stochastic variables determined by some probability distribution. We restrict our attention to nearest-neighbor transport in the limit in which the fluctuations in the hopping rates are large. In this limit we construct an exact expansion for the frequency-dependent diffusion coefficient D(\ensuremath{\varepsilon}) that is applicable to a wide range of transport phenomena (d=1 conductors, trapping phenomena, molecularly based electronic devices, etc.) in any spatial dimension. For the case of hopping transport with d=1, our method confirms earlier results that strong fluctuations in the hopping rates give rise to a non-Markovian ${\ensuremath{\varepsilon}}^{1/2}$ correction to normal diffusion. In two dimensions, we establish explicitly the existence of a non-Markovian logarithmic correction \ensuremath{\varepsilon} ln\ensuremath{\varepsilon} to D(\ensuremath{\varepsilon}). The formalism is then extended to d dimensions and the frequency corrections are discussed. For d=3, two frequency corrections must be retained. One is linear in \ensuremath{\varepsilon} and the other proportional to ${\ensuremath{\varepsilon}}^{3/2}$. It is shown that only the ${\ensuremath{\varepsilon}}^{3/2}$ correction contributes to the long-time tail ${t}^{\mathrm{\ensuremath{-}}3/2}$ in the time-dependent diffusion coefficient D(t). From these results we show that the long-time tail in the velocity autocorrelation function which is a consequence of the strong fluctuations in the hopping rates is of the form ${t}^{\mathrm{\ensuremath{-}}(1+d/2)}$. Comparison is made with earlier results.